Pentomino Crate: 12 Pieces into a 3 × 4 × 5 Box

Twelve chunky printed pentominoes that all have to fit into one tall 3 × 4 × 5 crate with a slide-in lid, a 3D packing puzzle with thousands of solutions.

Date: 2026-11-13 Category: Puzzles and Games Modeling difficulty: Intermediate Estimated print: ~20 h total · ~224 g PLA (estimate) Suggested price: $32–$55 printed · $6 digital file (estimate) Status: Concept

Description

Each of the twelve pieces is a flat pentomino (F, I, L, N, P, T, U, V, W, X, Y, Z), 16 mm thick. Together they are exactly 60 unit cubes, which is the volume of a 3 × 4 × 5 box. The puzzle is to pack them all in.

The crate is a tall box with an interior of 49 × 65 × 81 mm. A flat lid slides into rails along the top, so the box closes with the pieces inside and doubles as storage. Pieces are in three colours (deep blue, mustard, off-white) so a half-packed layer is easy to read.

Print time is long (about 20 hours), but every part is flat or a plain box, so none needs supports.

Why it sells

The 3 × 4 × 5 box is one of the three sizes in which the twelve pentacubes pack: Wikipedia lists 2 × 3 × 10 (12 solutions), 2 × 5 × 6 (264 solutions) and 3 × 4 × 5 (3,940 solutions) (Pentomino). Bouwkamp catalogued the 3 × 4 × 5 case in 1967 (Springer reference). Existing printable pentomino sets exist, such as a case puzzle on Printables (model 341492, page not opened) and a MakerWorld model (2393499, page not opened). I found no verified sales data. The puzzle shows demand as a classic and as an established printed category.

The gap: this design fixes the piece thickness at 16 mm with a measured 0.2 mm gap and ships a closed crate with rails rather than an open tray, so the packed set can be carried.

Who buys it

Puzzle collectors, maths teachers, gift buyers for adults who like logic puzzles. Etsy, craft fairs and Printables or MakerWorld for the file.

Design overview

• 12 flat pentominoes, each 16 mm thick, 5 cells of 16 mm: 60 cells in total.
• Crate interior 49 × 65 × 81 mm; the packed set is 47.8 × 63.8 × 79.8 mm, so 0.6 mm clearance per side.
• Rails: 2.2 mm deep, 3.4 mm high slots in the two long walls; the lid is 3 mm thick.
• Three colours spread across the twelve pieces so a half-packed layer reads clearly.
• No supports on any part; the 2.2 mm slot ceiling bridges.
• What sets it apart: a closed crate with a slide-in lid sized to the packed pieces, not an open tray.

Dimensions

Parts list

Hardware and non-printed parts

None — fully printed.

Fits and tolerances

• Lid in rails: sliding, 0.2 mm per side horizontally (53 mm wide in a 53.4 mm slot span) and 0.2 mm above and below (3.0 mm lid in a 3.4 mm slot).
• Pieces to each other: sliding, 0.1 mm per face offset, so 0.2 mm between neighbours. Looser than a press fit on purpose: pieces must go in one at a time.
• Packed set in cavity: 0.6 mm per side on every axis.

Modeling guide

Origin: crate, centre of the bottom face; +X right, +Y back, +Z up. Pieces: minimum corner of the piece.

Each of the 12 pieces.

1. Sketch on XY a 16 × 16 square at (16i, 16j) for each cell and merge them: F (1,0) (2,0) (0,1) (1,1) (1,2); I (0,0) to (4,0); L (0,0) (0,1) (0,2) (0,3) (1,3); N (0,0) (0,1) (1,1) (1,2) (1,3); P (0,0) (1,0) (0,1) (1,1) (0,2); T (0,0) (1,0) (2,0) (1,1) (1,2); U (0,0) (2,0) (0,1) (1,1) (2,1); V (0,0) (0,1) (0,2) (1,2) (2,2); W (0,0) (0,1) (1,1) (1,2) (2,2); X (1,0) (0,1) (1,1) (2,1) (1,2); Y (1,0) (0,1) (1,1) (1,2) (1,3); Z (0,0) (1,0) (1,1) (1,2) (2,2).
2. Offset the profile inward 0.1.
3. Extrude 15.8.
4. Chamfer the bottom perimeter 0.4 × 45°; fillet the top perimeter 0.5.

Crate.

1. Sketch on XY: rectangle 57 × 73 centered, corner radius 2. Extrude 90.
2. Sketch on the top face: rectangle 49 × 65 centered. Cut down 86.
3. Cut the front wall away above Z=85: box X −28.5 to 28.5, Y −36.5 to −32.5, Z 85 to 90.
4. Slots: on each long wall cut X ±24.5 to ±26.7, Z 85 to 88.4, Y −36.5 to +32.5.
5. Fillet the vertical outer edges 2; chamfer the bottom perimeter 0.4 × 45°.

Lid.

1. Rectangle 53 × 68.8 centered, corner radius 1. Extrude 3.
2. Cut a semicircle radius 6 centered on the front edge for a thumb notch.

Check: fit all twelve modeled pieces into the 3 × 4 × 5 cell volume without overlap; the 3,940 count is from the Wikipedia article.

Print settings

Assembly

1. Print twelve pieces, the crate and the lid.
2. Pack the twelve pieces into the crate, three across by four deep, five layers high.
3. Slide the lid into the rails from the front opening until it meets the back wall.

Variants and upsells

• 12 mm unit for a pocket crate; 20 mm for a display crate.
• A 2 × 5 × 6 crate (264 solutions) as a second, easier size.
• Challenge variant where one cell is pre-filled by a printed plug.

Selling and distribution

Etsy and craft fairs for the sets; Printables and MakerWorld for the file. Price from material (about 230 g) plus 20 hours of machine time plus margin. Personal-use file licence plus a small-commercial add-on. Keywords: pentomino puzzle, 3d printed puzzle, 3x4x5 box puzzle, packing puzzle, polycube, logic puzzle gift, maths gift, STEM puzzle, brain teaser, wooden puzzle alternative.

Risks and IP notes

• Very long print; mitigation: print the pieces in colour batches and offer a smaller unit size.
• Existing free pentomino files; mitigation: sell the crate and finish.
• Tight fits bind; mitigation: 0.2 mm gaps and a test piece first.
• Count of solutions is cited from one reference; mitigation: don't put it in listings as a guarantee.

Resource links

• Pentomino (Wikipedia) — Box sizes and 3,940 solutions for 3x4x5
• Solid pentominoes (puzzler docs) — Solid pentomino puzzle background
• Pentomino case puzzle (Printables) — Existing printable pentacube set with a storage case; page not opened
• Pentomino model (MakerWorld) — Existing printable pentomino model; page not opened
• Pentominoes (Mathematische Basteleien) — Piece shapes
• Bouwkamp catalogue reference (Springer) — Catalogue of 3x4x5 solutions, 1967